List of top Mathematics Questions on Quadratic Equations asked in TS EAMCET

Two real roots of \[ 3x^4+ax^3+55x^2-52x+12=0 \] are positive and equal. Product of other two roots is 1. If roots belong to natural numbers then \[ a\beta-a+\frac{\gamma}{\delta}= \]

If \( \alpha, \beta, \gamma, \delta \) are the roots of the equation \( x^4 - 10x^3 + 35x^2 - 50x + 24 = 0 \) and \( \alpha^2, \beta^2, \gamma^2, \delta^2 \) are the roots of the equation \( x^4 + ax^3 + bx^2 + cx + d = 0 \), then \( a = \)

If \( \alpha, \beta \in \mathbb{R} \) and \( \alpha \leq \frac{x^2 - 3x + 4}{x^2 - 4x + 5} \leq \beta \) for all \( x \in \mathbb{R} \), then \( (2\alpha - 3)^2 + (2\beta - 3)^2 = \)

If two roots of the equation \( x^5 - 9x^4 + 27x^3 - 23x^2 - 24x + 36 = 0 \) are repeated roots of multiplicity 2 and all the roots are integers then the sum of the squares of all different roots of the equation is:

If all roots of \[ x^5-3x^4+2x^3-3x^2+5x-2=0 \] are increased by real value h so that term containing \(x^3\) vanishes in transformed equation and h is integer, then h=

If \(\alpha\in R\) and equation \[ (x-\alpha)(x-3)+1=0 \] has equal roots, then sum of squares of all values of \(\alpha\) is

If \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 - Px^2 + Qx - R = 0 \) and \( (\alpha-2)^2, (\beta-2)^2, (\gamma-2)^2 \) are the roots of the equation \( x^3 - 5x^2 + 4x = 0 \), then the possible least value of \( P+Q+R \) is

The number of all common roots of the equation \( x^4 - 10x^3 + 37x^2 - 60x + 36 = 0 \) and the transformed equation of it obtained by increasing any two distinct roots of it by 1, keeping the other two roots fixed, is

If the equation \( x^2 - 3ax + a^2 - 2a - K = 0 \) has different real roots for every rational number \( a \), then \( K \) lies in the interval

Find the value of \( k \) such that \( 5 \cdot 2^n - 48n + k \) is divisible by 24.

The solution set of the inequality \( \sqrt{x^2 + x - 2}>(1 - x) \) is:

The solution set of the inequality \[ 3^x + 3^{1-x} - 4 <0 \] contained in \( \mathbb{R} \) is:

If \( \alpha, \beta \) are the real roots of the equation \( x^2 + ax + b = 0 \). If \( \alpha + \beta = \frac{1}{2} \) and \( \alpha^3 + \beta^3 = \frac{37}{8} \), then \( {a}-{\frac{1}{b}} =\)

$$ x^5 + 4x^4 - 13x^3 - 52x^2 + 36x + 144 = 0 $$

The common solution set of the inequalities \[ x^2 - 4x \leq 12 \quad \text{and} \quad x^2 - 2x \geq 15 \] taken together is:

If \( \alpha, \beta, \gamma \) are the roots of the equation \( 4x^3 - 3x^2 + 2x - 1 = 0 \), then \( \alpha^3 + \beta^3 + \gamma^3 = \)

The equation \( 16x^4 + 16x^3 - 4x - 1 = 0 \) has a multiple root. If \( \alpha, \beta, \gamma, \delta \) are the roots of this equation, then \( \frac{1}{\alpha^4} + \frac{1}{\beta^4} + \frac{1}{\gamma^4} + \frac{1}{\delta^4} = \)

If are the roots of the equation \[ 2x^3 \;-\; 5x^2 \;+\; 4x \;-\; 3 \;=\; 0, \] then \[ \sum \alpha\beta(\alpha+\beta) \;=\;? \]